The many laws of large numbers

The statistical average, I wrote in Chapter 1 of Numbers Rule Your World, is the one invention that statisticians have inserted into our popular lexicon. To a lesser extent, the "law of large numbers" is another statistical concept that has attracted common usage, especially among the business community.

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The other day, I was making copies at Fedex Kinkos, and the "law of large numbers" suddenly loomed large. Scratch that -- I was paying for my copies, all 178 black-and-white letter-sized pages.

No, I did not use any 11x17 paper. As I have made a lot of copies on this one machine over the years, I can tell you this is not the first time this mistake has appeared.

And this led me to think: 22 cents is peanuts, most customers would ignore the error (unless it happens to you many times), and by the law of large numbers, a little extra from a lot of customers equal a lot of revenues, illicitly obtained.

An estimate: 1800 stores, 2 machines per store, 50 customers per machine per day, 300 days per year would add up to 54 million transactions per year. Sneak in 22 cents once every 10 transactions would yield $1.2 million.

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Except, I just mis-applied the law of large numbers, as conceived by statisticians. The real law of large numbers says that the more data we have, the more likely the average value of the observed data is representative. (There is a pre-condition, the data has to be "independent and identically distributed"... a random sample would qualify.) In practical terms, this is the reason why we can poll 1000 people and then make general statements about the population.

That's the real law. When commonly used, the "law of large numbers" takes on many meanings. I did a search on Google and found the following "laws":

the law of large numbers dictates that Apple’s growth prospects must
decline sharply in the years ahead; otherwise, it would soon grow
larger than the U.S. economy itself! (Investing Daily) [the fallacy of linear extrapolation?]

Those 23 banks still under Fed supervision would control an enormous
amount of money, of course. The "law of large numbers" dictates that
those are the banks with the most influence over the economy. (Huffington Post) [the definition of large?]

Still, Google's growth has slowed in recent years. Sure, part of that
is simply the so-called law of large numbers, i.e. the bigger a company
gets, the more difficult it is to keep posting gaudy percentage
increases in sales and earnings. (CNN Money) [the law of diminishing returns? diseconomies of scale?]

The law of large numbers has to catch up with China just as it did with
the other miracle economies, including Japan and South Korea during
their industrialization phase. (Newsweek) [Me: the law of diminishing returns? regression to the mean?]

Most of the above are valid laws, they describe what could happen when things grow large -- influence increases, returns diminish, untapped market diminishes, etc. but none of the above has to do with the statistical "law of large numbers".

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Here is a correct use of the law:

If you run a bunch of casinos with hundreds of thousands of punters coming
and and betting hundreds of millions of dollars, then you can predict with some
accuracy the amount of money you're going to make at the end of the quarter:
it's called the law of large numbers. (Felix Salmon)

With this sort of sample size, "some accuracy" can be replaced by "high accuracy". In this sentence, we can see repeated observations (bets) used to predict (generalize about) the average gain/loss.